Weighted L^p‐theory for vector potential operators in three‐dimensional exterior domains

Abstract

In the present paper, we study the vector potential problem in exterior domains of R 3 . Our approach is based on the use of weighted spaces in order to describe the behavior of functions at infinity. As a first step of the investigation, we prove important results on the Laplace equation in exterior domains with Dirichlet or Neumann boundary conditions. As a consequence of the obtained results on the vector potential problem, we establish useful results on weighted Sobolev inequalities and Helmholtz decompositions of weighted spaces.

Weighted Hilbert spaces for the stationary exterior Stokes problem with Navier slip boundary conditions

“…Here, the arguments are based on adequate extensions of solutions to (1.4)-(1.5) that we already have at hand. These techniques were previoulsy applied in [33] (see also [42]) for the exterior Laplace equation with Neumann boundary condition.…”

$L^p$-theory for the exterior Stokes problem with Navier's type slip-without-friction boundary conditions

“…Our first proposition is established also in [24], it characterizes the kernel of the Laplace operator with Neumann boundary condition. For any integer k ∈ Z and 1 < p < ∞, Here also, we set N ∆ p,α = {0} when α ≤ 0; N ∆ p,α is a finite-dimentional space of the same dimension as P ∆ [1−3/p−α] and in particular,…”

“…Giroire in [21], studied the problem (2.20) in the Hilbert framework. In [24], the authors investigated the harmonic Neumann problem in L p theory, for the exterior domain with boundary of class C 1,1 , note that these results have been also proved by Specovius-Neugebauer [32] with boundary of class at leas C 2 . C. Amrouche, V. Girault and J. Giroire in [3], studied the problem (2.20) with data f belongs to W −1,p 0…”

“…We shall precise in which sense, the Navier's type slip-without-friction boundary conditions (1.3), we finish this section by giving a result concerning the Laplace problem with Neumann boundary conditions with data f belongs to L p (Ω), we got the existence of solutions in W 1,p −1 (Ω). In Section 3, we recall one result about the vector potential problem (see [24]) and we prove the Inf-Sup conditions, which plays a crucial role in the existence and uniqueness of solutions. Finally, in Section 4, we conclude with the main result of this paper related to well-posedness of the Stokes problem (1.1)- (1.3).…”

Very weak solution for the exterior stationary Stokes equations with Navier slip boundary condition

“…Then f belongs to L 2 (R 3 ). Thanks [6, Theorem 3.9], there exists a unique function v ∈ W 2,2 0 (R 3 ) such that [29,Theorem 2.7], that the following problem:…”